signal & system lab - electronics club · operations on signals and sequences such as addition,...

0 ShriRam College of Engineering & Managment

National Expressway (A.B. Road), Banmore

SIGNAL & SYSTEM LAB CONTENTS

S. No. List of Experiment Page

No

1. Generate a matrix and perform basic operation on matrices using MATLAB

software. 01-03

2.

Generation of Various Signals and Sequences (Periodic and Aperiodic), such

as Unit Impulse, Unit Step, Square, Saw tooth, Triangular, Sinusoidal,

Ramp, Sinc.

04-06

3. Operations on Signals and Sequences such as Addition, Multiplication,

Scaling, Shifting, Folding, Computation of Energy and Average Power 07-10

4. Finding the Even and Odd parts of Signal/Sequence and Real and Imaginary

parts of Signal 11-13

5. Convolution between Signals and sequences 14-15

6. Auto Correlation and Cross Correlation between Signals and Sequences. 16-18

7.

Computation of Unit sample, Unit step and Sinusoidal responses of the

given LTI system and verifying its physical realiazability and stability

properties.

19-21

8.

To find the Fourier Transform of a given signal and plotting its magnitude and

phase spectrum.

22-23

Lab In-charge



Experiment No.1

Objective:

Generate a matrix and perform basic operation on matrices using MATLAB software. Software Required: MATLAB software

Theory: -

MATLAB treats all variables as matrices. Vectors are special forms of matrices and contain

only one row or one column. Whereas scalars are special forms of matrices and contain only

one row and one column. A matrix with one row is called row vector and a matrix with single

column is called column vector.

The first one consists of convenient matrix building functions, some of which are given

below.

1. eye - identity matrix

2. zeros - matrix of zeros

3. ones - matrix of ones

4. diag - extract diagonal of a matrix or create diagonal matrices

5. triu - upper triangular part of a matrix

6. tril - lower triangular part of a matrix

7. rand - ran

commands in the second sub-category of matrix functions are

1. size- size of a matrix

2. det -determinant of a square matrix

3. inv- inverse of a matrix

4. rank- rank of a matrix

5. rref- reduced row echelon form

6. eig- eigenvalues and eigenvectors

7. poly- characteristic polynomialdomly generated matrix

Program:

% Creating a column vector

>> a=[1;2;3]

a =

1

2

3

% Creating a row vector >> b=[1 2 3] % Creating a matrix

>> m=[1 2 3;4 6 9;2 6 9]

m =

1 2 3

4 6 9

2 6 9

% Extracting sub matrix from matrix

>> sub_m=m(2:3,2:3)

sub_m =



6 9

6 9

% extracting column vector from matrix

>> c=m (:,2)

c =

2

6

6

% extracting row vector from matrix

>> d=m (3,:)

d =

2 6 9

% creation of two matrices a and b

>> a=[2 4 -1;-2 1 9;-1 -1 0]

a =

2 4 -1

-2 1 9

-1 -1 0

>> b=[0 2 3;1 0 2;1 4 6]

b =

0 2 3

1 0 2 1 4 6 % matrix multiplication

>> x1=a*b

x1 =

3 0 8

10 32 50

-1 -2 -5

% element to element multiplication

>> x2=a.*b

x2 =

0 8 -3

-2 0 18

-1 -4 0

% matrix addition

>> x3=a+b

x3 =

2 6 2

-1 1 11

0 3 6

% matrix subtraction

>> x4=a-b

x4 =

2 2 -4

-3 1 7

-2 -5 -6

% matrix division

>> x5=a/b



x5 =

-9.0000 -3.5000 5.5000

12.0000 3.7500 -5.7500

3.0000 0.7500 -1.7500

% element to element division

>> x6=a./b Warning: Divide by zero. x6 =

Inf 2.0000 -0.3333

-2.0000 Inf 4.5000

-1.0000 -0.2500 0

% inverse of matrix a

>> x7=inv(a)

x7 =

-0.4286 -0.0476 -1.7619

0.4286 0.0476 0.7619

-0.1429 0.0952 -0.4762

% transpose of matrix a

>> x8=a'

x8 =

2 -2 -1

4 1 -1

-1 9 0

RESULT: Matrix operations are performed using Matlab software.

-------------------------------



Experiment No.2

Object:

Generate various signals and sequences (Periodic and aperiodic), such as Unit Impulse,

Unit Step, Square, Saw tooth, Triangular, Sinusoidal, Ramp, Sinc.

Software Required: Matlab software

Theory: If the amplitude of the signal is defined at every instant of time then it is called

continuous time signal. If the amplitude of the signal is defined at only at some instants

of time then it is called discrete time signal. If the signal repeats itself at regular

intervals then it is called periodic signal. Otherwise they are called aperiodic signals.

EX: ramp,Impulse,unit step, sinc- Aperiodic signals square,sawtooth,triangular

sinusoidal – periodic signals.

Ramp sinal: The ramp function is a unitary real function, easily computable as the

mean of the independent variable and its absolute value.This function is applied in

engineering. The name ramp function is derived from the appearance of its graph.

Unit impulse signal: One of the more useful functions in the study of linear systems is

the "unit impulse function."An ideal impulse function is a function that is zero

everywhere but at the origin, where it isinfinitely high. However, the area of the

impulse is finite.

Unit step signal: The unit step function and the impulse function are considered to be

fundamental functions in engineering, and it is strongly recommended that the reader

becomes very familiar with both of these functions.

Sinc signal: There is a particular form that appears so frequently in communications

engineering, that we give it its own name. This function is called the "Sinc function”.

The Sinc function is defined in the following manner:

The value of sinc(x) is defined as 1 at x = 0, since



Procedure:

Open MATLAB

Open new M-file

Type the program

Save in current directory

Compile and Run the program For the output see command window\ Figure window

PROGRAM: % Generation of signals and sequences

clc;

clear all;

close all;

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of unit impulse signal

t1=-1:0.01:1

y1=(t1==0);

subplot(2,2,1);

plot(t1,y1);

xlabel('time');

ylabel('amplitude');

title('unit impulse signal');

%generation of impulse sequence

subplot(2,2,2);

stem(t1,y1);

xlabel('n');


title('unit impulse sequence');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of unit step signal

t2=-10:1:10;

y2=(t2>=0);

subplot(2,2,3);

plot(t2,y2);

xlabel('time');


title('unit step signal');

%generation of unit step sequence

subplot(2,2,4);

stem(t2,y2);

xlabel('n');


title('unit step sequence');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of square wave signal

t=0:0.002:0.1; y3=square(2*pi*50*t);

figure;



subplot(2,2,1);

plot(t,y3);

axis([0 0.1 -2 2]);

xlabel('time');


title('square wave signal');

%generation of square wave sequence

subplot(2,2,2);

stem(t,y3);

axis([0 0.1 -2 2]);

xlabel('n');


title('square wave sequence');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of sawtooth signal

y4=sawtooth(2*pi*50*t);

subplot(2,2,3);

plot(t,y4);

axis([0 0.1 -2 2]);

xlabel('time');


title('sawtooth wave signal');

%generation of sawtooth sequence

subplot(2,2,4);

stem(t,y4);

axis([0 0.1 -2 2]);

xlabel('n');


title('sawtooth wave sequence');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of triangular wave signal

y5=sawtooth(2*pi*50*t,.5);

figure;

subplot(2,2,1);

plot(t,y5);

axis([0 0.1 -2 2]);

xlabel('time');


title(' triangular wave signal');

%generation of triangular wave sequence

subplot(2,2,2);

stem(t,y5);

axis([0 0.1 -2 2]);

xlabel('n');


title('triangular wave sequence'); %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of sinsoidal wave signal

y6=sin(2*pi*40*t);

subplot(2,2,3);



plot(t,y6);

axis([0 0.1 -2 2]);

xlabel('time');


title(' sinsoidal wave signal');

%generation of sin wave sequence

subplot(2,2,4);

stem(t,y6);

axis([0 0.1 -2 2]);

xlabel('n');


title('sin wave sequence');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of ramp signal

y7=t;

figure;

subplot(2,2,1);

plot(t,y7);

xlabel('time');


title('ramp signal');

%generation of ramp sequence

subplot(2,2,2);

stem(t,y7);

xlabel('n');


title('ramp sequence');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%generation of sinc signal

t3=linspace(-5,5);

y8=sinc(t3);

subplot(2,2,3);

plot(t3,y8);

xlabel('time');


title(' sinc signal');

%generation of sinc sequence

subplot(2,2,4);

stem(y8);

xlabel('n');

ylabel('amplitude'); title('sinc sequence');

Result: Various signals & sequences generated using Matlab software.



output:



------------------------------



Experiment No.3

Object: perform the operations on signals and sequences such as addition,

multiplication, scaling, shifting, folding and also compute energy and power.

Software Required: Matlab software.

Theory:

Signal Addition

Addition: any two signals can be added to form a third signal,

z (t) = x (t) + y (t)

Multiplication :

Multiplication of two signals can be obtained by multiplying their values at every

instants . z

z(t) = x (t) y (t)

Time reversal/Folding:

Time reversal of a signal x(t) can be obtained by folding the signal about t=0.

Y(t)=y(-t)

Signal Amplification/Scaling : Y(n)=ax(n) if a < 1 attnuation

a >1 amplification

Time shifting: The time shifting of x(n) obtained by delay or advance the signal in time

by

using y(n)=x(n+k)

If k is a positive number, y(n) shifted to the right i e the shifting delays the signal

If k is a negative number, y(n ) it gets shifted left. Signal Shifting advances the signal

Energy:



Average power:

Program:

clc;

clear all;

close all;

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

% generating two input signals

t=0:.01:1;

x1=sin(2*pi*4*t);

x2=sin(2*pi*8*t);

subplot(2,2,1);

plot(t,x1);

xlabel('time');


title('input signal 1');

subplot(2,2,2);

plot(t,x2);

xlabel('time');


title('input signal 2');

% addition of signals

y1=x1+x2;

subplot(2,2,3);

plot(t,y1);

xlabel('time');


title('addition of two signals');

% multiplication of signals

y2=x1.*x2;

subplot(2,2,4);

plot(t,y2);

xlabel('time');


title('multiplication of two signals');

% scaling of a signal1

A=2;

y3=A*x1;

figure;

subplot(2,2,1);

plot(t,x1);

xlabel('time');




title('input signal')

subplot(2,2,2);

plot(t,y3);

xlabel('time');


title('amplified input signal');

% folding of a signal1

h=length(x1);

nx=0:h-1;

subplot(2,2,3);

plot(nx,x1);

xlabel('nx');



y4=fliplr(x1);

nf=-fliplr(nx);

subplot(2,2,4);

plot(nf,y4);

xlabel('nf');


title('folded signal');

%shifting of a signal 1

figure;

subplot(3,1,1);

plot(t,x1);

xlabel('time t');


title('input signal');

subplot(3,1,2);

plot(t+2,x1);

xlabel('t+2');


title('right shifted signal');

subplot(3,1,3);

plot(t-2,x1);

xlabel('t-2');


title('left shifted signal');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

%operations on sequences

n1=1:1:9;

s1=[1 2 3 0 5 8 0 2 4];

figure;

subplot(2,2,1);

stem(n1,s1);

xlabel('n1');


title('input sequence1');

s2=[1 1 2 4 6 0 5 3 6];



subplot(2,2,2);

stem(n1,s2);

xlabel('n2');


title('input sequence2');

% addition of sequences

s3=s1+s2;

subplot(2,2,3);

stem(n1,s3);

xlabel('n1');


title('sum of two sequences');

% multiplication of sequences

s4=s1.*s2;

subplot(2,2,4);

stem(n1,s4);

xlabel('n1');


title('product of two sequences');

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

% program for energy of a sequence

z1=input('enter the input sequence');

e1=sum(abs(z1).^2);

disp('energy of given sequence is');e1

% program for energy of a signal

t=0:pi:10*pi;

z2=cos(2*pi*50*t).^2;

e2=sum(abs(z2).^2);

disp('energy of given signal is');e2

% program for power of a sequence

p1= (sum(abs(z1).^2))/length(z1);

disp('power of given sequence is');p1

% program for power of a signal

p2=(sum(abs(z2).^2))/length(z2);

disp('power of given signal is');p2

OUTPUT:

enter the input sequence[1 3 2 4 1]

energy of given sequence is

e1 = 31

energy of given signal is

e2 = 4.0388

power of given sequence is

p1 = 6.2000

power of given signal is

p2 = 0.3672

Result: Various operations on signals and sequences are performed.



Output:



-------------------------------



Experiment No.4

Object: - Finding even and odd part of the signal and sequence and also find real and imaginary

parts of signal.


Theory: One of characteristics of signal is symmetry that may be useful for signal analysis.

Even signals are symmetric around vertical axis, and Odd signals are symmetric about origin.

Even Signal: A signal is referred to as an even if it is identical to its time-reversed

counterparts; x(t) = x(-t).

Odd Signal: A signal is odd if x(t) = -x(-t).

An odd signal must be 0 at t=0, in other words, odd signal passes the origin.

Using the definition of even and odd signal, any signal may be decomposed into a sum of

its even part, xe(t), and its odd part, xo(t), as follows

Even and odd part of a signal: Any signal x(t) can be expressed as sum of even and odd

components i.e .,

x(t)=xe(t)+xo(t)

Program:

clc

close all;

clear all;

%Even and odd parts of a signal

t=0:.001:4*pi;

x=sin(t)+cos(t); % x(t)=sint(t)+cos(t)

subplot(2,2,1)

plot(t,x)

xlabel('t');

ylabel('amplitude')


y=sin(-t)+cos(-t); % y(t)=x(-t)

subplot(2,2,2)

plot(t,y)

xlabel('t');

ylabel('amplitude')

title('input signal with t= -t')

even=(x+y)/2;

subplot(2,2,3)

plot(t,even)

xlabel('t');



ylabel('amplitude')

title('even part of the signal')

odd=(x-y)/2;

subplot(2,2,4)

plot(t,odd)

xlabel('t');


title('odd part of the signal');

% Even and odd parts of a sequence

x1=[0,2,-3,5,-2,-1,6];

n=-3:3;

y1= fliplr(x1);%y1(n)=x1(-n)

figure;

subplot(2,2,1);

stem(n,x1);

xlabel('n');


title('input sequence');

subplot(2,2,2);

stem(n,y1);

xlabel('n');


title('input sequence with n= -n');

even1=.5*(x1+y1);

odd1=.5*(x1-y1);

% plotting even and odd parts of the sequence

subplot(2,2,3);

stem(n,even1);

xlabel('n');


title('even part of sequence');

subplot(2,2,4);

stem(n,odd1);

xlabel('n');


title('odd part of sequence');

% plotting real and imginary parts of the signal

x2=sin(t)+j*cos(t);

figure;

subplot(3,1,1);

plot(t,x2);

xlabel('t');



subplot(3,1,2)

plot(t,real(x2));

xlabel('time');


title('real part of signal');

subplot(3,1,3)

plot(t,imag(x2));

xlabel('time');


title('imaginary part of siganl');



RESULT: Even and odd part of the signal and sequence, real and imaginary parts of signal

are computed. Output:



-------------------------------



Experiment No.5

Object: - Write the program for convolution between two signals and also between two

sequences.


Theory:-

Convolution involves the following operations.

1. Folding

2. Multiplication

3. Addition

4. Shifting

These operations can be represented by a Mathematical Expression as follows:

x[n]= Input signal Samples

h[ n-k]= Impulse response co-efficient.

y[ n]= Convolution output.

n = No. of Input samples

h = No. of Impulse response co-efficient.

Example : X(n)={1 2 -1 0 1}, h(n)={ 1,2,3,-1}

Program: clc;

close all;

clear all;

%program for convolution of two sequences

x=input('enter input sequence: ');

h=input('enter impulse response: ');

y=conv(x,h);

subplot(3,1,1);

stem(x);

xlabel('n');

ylabel('x(n)');

title('input sequence')

subplot(3,1,2);

stem(h);

xlabel('n');

ylabel('h(n)');

title('impulse response sequence')

subplot(3,1,3);

stem(y);



xlabel('n');

ylabel('y(n)');

title('linear convolution')

disp('linear convolution y=');

disp(y)

%program for signal convolution

t=0:0.1:10;

x1=sin(2*pi*t);

h1=cos(2*pi*t);

y1=conv(x1,h1);

figure;

subplot(3,1,1);

plot(x1);

xlabel('t');

ylabel('x(t)');


subplot(3,1,2);

plot(h1);

xlabel('t');

ylabel('h(t)');

title('impulse response')

subplot(3,1,3);

plot(y1);

xlabel('n');

ylabel('y(n)');

title('linear convolution');

RESULT: convolution between signals and sequences is computed.

Output:

enter input sequence: [1 3 4 5]

enter impulse response: [2 1 4]

linear convolution y=

2 7 15 26 21 20



Experiment No.6

Objective:

To compute Auto correlation and Cross correlation between signals and sequences.

Software Required: Mat lab software

Theory:

Correlations of sequences:

It is a measure of the degree to which two sequences are similar. Given two real-valued

sequences x(n) and y(n) of finite energy,

Convolution involves the following operations.

1. Shifting

2. Multiplication

3. Addition

These operations can be represented by a Mathematical Expression as follows:

Cross correlation

The index l is called the shift or lag parameter

Autocorrelation

Program:

clc;

close all;

clear all;

% two input sequences

x=input('enter input sequence');

h=input('enter the impulse suquence');

subplot(2,2,1);

stem(x);

xlabel('n');

ylabel('x(n)');

title('input sequence');

subplot(2,2,2);

stem(h);

xlabel('n');

ylabel('h(n)');

title('impulse sequence');

% cross correlation between two sequences



y=xcorr(x,h);

subplot(2,2,3);

stem(y);

xlabel('n');

ylabel('y(n)');

title(' cross correlation between two sequences ');

% auto correlation of input sequence

z=xcorr(x,x);

subplot(2,2,4);

stem(z);

xlabel('n');

ylabel('z(n)');

title('auto correlation of input sequence');

% cross correlation between two signals

% generating two input signals

t=0:0.2:10;

x1=3*exp(-2*t);

h1=exp(t);

figure;

subplot(2,2,1);

plot(t,x1);

xlabel('t');

ylabel('x1(t)');


subplot(2,2,2);

plot(t,h1);

xlabel('t');

ylabel('h1(t)');

title('impulse signal');

% cross correlation

subplot(2,2,3);

z1=xcorr(x1,h1);

plot(z1);

xlabel('t');

ylabel('z1(t)');

title('cross correlation ');

% auto correlation

subplot(2,2,4);

z2=xcorr(x1,x1);

plot(z2);

xlabel('t');

ylabel('z2(t)');

title('auto correlation ');

Result: Auto correlation and Cross correlation between signals and sequences is computed.

Output: enter input sequence [1 2 5 7]

enter the impulse sequence [2 6 0 5 3]



-------------------------------



Experiment No.7 Objective:

Compute the Unit sample, unit step and sinusoidal response of the given LTI system

and verifying its stability.

Software Required: Mat lab software 7.0 and above

Theory:

A discrete time system performs an operation on an input signal based on predefined criteria

to produce a modified output signal. The input signal x(n) is the system excitation, and y(n) is

the system response. The transform operation is shown as,

If the input to the system is unit impulse i.e. x(n) = δ(n) then the output of the system is

known as impulse response denoted by h(n) where,

h(n) = T[δ(n)]

we know that any arbitrary sequence x(n) can be represented as a weighted sum of discrete

impulses. Now the system response is given by,

For linear system (1) reduces to

%given difference equation y(n)-y(n-1)+.9y(n-2)=x(n);



Program: %given difference equation y(n)-y(n-1)+.9y(n-2)=x(n);

clc;

clear all;

close all;

b=[1];

a=[1,-1,.9];

n =0:3:100;

%generating impulse signal

x1=(n==0);

%impulse response

y1=filter(b,a,x1);

subplot(3,1,1);

stem(n,y1);

xlabel('n');

ylabel('y1(n)');

title('impulse response');

%generating step signal

x2=(n>0);

% step response

y2=filter(b,a,x2);

subplot(3,1,2);

stem(n,y2);

xlabel('n');

ylabel('y2(n)')

title('step response');

%generating sinusoidal signal

t=0:0.1:2*pi;

x3=sin(t);

% sinusoidal response

y3=filter(b,a,x3);

subplot(3,1,3);

stem(t,y3);

xlabel('n');

ylabel('y3(n)');

title('sin response');

% verifing stability

figure;

zplane(b,a);

Result: The Unit sample, unit step and sinusoidal response of the given LTI system is

computed and its stability verified. Hence all the poles lie inside the unit circle, so system is

stable.



Output:

-------------------------------



Experiment No.8

Object: To find the Fourier Transform of a given signal and plotting its magnitude and phase

spectrum.


Theory:

Fourier Transform:

The Fourier transform as follows. Suppose that ƒ is a function which is zero outside of some

interval [−L/2, L/2]. Then for any T ≥ L we may expand ƒ in a Fourier series on the interval

[−T/2,T/2], where the "amount" of the wave e2πinx/T in the Fourier series of ƒ is given by

By definition Fourier Transform of signal f(t) is defined as

Inverse Fourier Transform of signal F(w) is defined as

Program: clc;

clear all;

close all;

fs=1000;

N=1024; % length of fft sequence

t=[0:N-1]*(1/fs);

% input signal

x=0.8*cos(2*pi*100*t);

subplot(3,1,1);

plot(t,x);

axis([0 0.05 -1 1]);

grid;

xlabel('t');



% Fourier transformof given signal

x1=fft(x);

% magnitude spectrum

k=0:N-1;

Xmag=abs(x1);

subplot(3,1,2);

plot(k,Xmag);

grid;

xlabel('t');


title('magnitude of fft signal')

%phase spectrum

Xphase=angle(x1);

subplot(3,1,3);



plot(k,Xphase);

grid;

xlabel('t');

ylabel('angle');

title('phase of fft signal');

Result: Magnitude and phase spectrum of FFT of a given signal is plotted.

Output:

-------------------------------

signal & system lab - electronics club · operations on signals and sequences such as addition,...

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